Here are some improvisations in various
tunings. Notation: tet stands for "tone equal temperament".
E.g. 55-tet is a scale with 55 equally spaced notes per octave.
If new to the use of ratios and cents to describe a scale, have a
look over the newbie notes for the Tunes page.

These pieces are inspired by the various
tunings - the feel of them when improvising. I find this is quite
marked, when playing, but notice it far less when listening,
especially for the subtler temperaments (temperament = the exact
tuning used for the notes of the scale, and the sharps and flats).

In the nineteenth century and earlier,
listening out for the qualities of a temperament must have been
something musical listeners were quite accustomed to, as the keys
varied in their tuning depending on how near or remote one was
from C major. However nowadays we aren't that used to this way of
listening to Western diatonic / twelve tone music, as all the
keys sound the same in the standard 12-tet tuning of modern
pianos.

At any rate, it is interesting to hear
them in other tunings, such as 12-tet. I think the inspiration of
the original tuning carries through when one does this, or can do.

Here is a truly glorious realisation of
one of my pieces in 12-tet by Mary Ackerley - Thanks Mary :-).

While improvising on this occasion and in
this timbre, I felt that the major thirds were the consonances to
resolve to and the fourths were comparative dissonances. The
third in this scale is very pure, within half a cent of 5/4.

This is in the mode 0 2 3 6 7 9 12 13 of
the scale with 13 equal divisions of 32/3. It's the same mode as
the next improvisation, but all the steps are three and a half
times larger (approx).

So, really huge steps - the "semi-tone"
in terms of fingering is a minor third, and it's a non octave
scale repeating at 32/3. The thirteenth root of 32/3 happens to
be within half a cent of a just intonation minor third at 6/5.

This is a very exotic tuning as it has no
pure fifths. This minor mode has the fifth more than a quarter
tone flat and the fourth more than a quarter tone sharp.

In terms of degrees of 13-tet it is:

0 2 3 6 7 9 12 13

As twelve tone equal note names:

C D- Eb-" F+++' G---" Ab++ B' C

where - = 2/13th of a tone flat, + = 2/13th
of a tone sharp, and ' = 1/13th of a tone sharp " = 1/13th
of a tone flat. I've written the middle two notes as F+++' and G---"
instead of F#--' and F#++" because in the mode they are
playing the role of F and G, if one thinks of it as a kind of
minor scale.

In cents, its

0.0 184.615 276.923
553.846 646.154 830.769 1107.69 1200.0

Such tunings only sound sweet if you
choose the timbre carefully. This one uses the Sitar voice of my
SB live! S/w Synth,. which is the one used for the original
improvisation, and happens to work well.

The mode includes the diminished 7th 0 3
6 9 13, which is interesting in 13-tet as it is made from
stacking three minor thirds at 276.92 cents and a major third at
369.23 cents, instead of the four minor thirds that one is used
to in 12-tet.

If you stack seven 13-tet "minor
thirds" on top of each other,

0 3 6 9 12 15 18

then reduce into the octave, you get 0 2
3 5 6 9 12 13, which differs from this scale by a single note. It
has the three note diminished seventh in five places.

0 2 3 6 7 9 12 13 itself consists of a
six note 13-tet diminished seventh 0 3 6 9 12 15 and one other
note, the flattened "fifth" at 7, and has the three
note diminished seventh in four places.

I recorded it using the Sitar voice of
the SB LIve! soundcard Creative S/W synth.

13-tet has no pure fifths at all, nearest
notes is at 738.46 cents, a third tone out, and this
improvisation uses instead a "fifth" of 646.154 cents,
more than a quarter tone flat. However, if you choose the right
timbre, 13-tet is a great scale. It is very timbre specific, what
sounds sweet on one timbre sounds raucous on another.

I'm not sure why it sounds so nice on the
SB LIve! sitar. Even on the SB live, works on the Creative S/W
synth and not on the SB Live! Midi Synth 8 Mb soundbank.

One can experiment to see which voice it
sounds best on. It sounds pretty strange on (say) the piano
timbre.

Improvisation in 24-tet

This explores use of the "inconsistency"
of 24-tet - not really the scale itself that is inconsistent, but
if you try approximating the intervals 5/4 and 7/4 in 24-tet, you
find that the approximations for 5/4 and 7/5 don't add up to the
one for 7/4. So, assumption that one can add approximations for
intervals and get the approximation for the multiple of the two
is inconsistent for this scale.

This can be used to modulate up and down
by a quarter tone, which I'm doing here in this short melody.

Go up by closest approximation to 7/4,
then down by closest approx to 5/7, and you end up at 5/4 plus a
quarter tone. Do this twice and you can modulate from C to C#.

Exactly that happens in this short
example melody - and at the end it drops down to the original C
so you can hear how remote it has become by then.

In the opening dom7th chord, notice how
the 7/5 interval from the major third to the seventh is a bit
"sour". A little later the same interval is played with
the third a quarter tone sharp, making it sweeter.

" The final, intriguing tuning goes
all the way to a 13-limit (see Table). This includes a complete
eight-tone harmonic scale on the 4/3 and a seven-tone one on 1/1.
Such scales sound fascinating in their variety, coherence, and
newness. However, the other tonalities (except 3/2 minor) are
pretty strange."

The range is perfect for a 'cello (also a
favourite instrument of mine). I've no idea how practical it
would be to play or how many 'cellos would actually be needed if
it was. Two anyway as it has a couple of six note chords in it;
perhaps three at times would really be needed.

This uses the 55-tet major scale, plus
accidentals.

This is a most delightful major scale (I
find). The fifth is just a bit unstable, at 698.182 cents, with a
nice major third at 392.727 cents - just a little sharp on 5/4.

As the accidentals I'm using C#, F#, and
G# at + 3 steps, and Eb, and Bb = -4 steps.

So as a twelve tone scale it's: 3 6 5 4 5
3 6 3 6 5 4 5.

This scale has extra-sharp major thirds
are at 436.364 cents (e.g. steps 6 3 6 5 = 20), and a couple of
narrow major thirds at 370.9 cents (steps 5 4 5 3 = 17), and most
of the major thirds are at 18 steps = 392.727 cents.

With these new sharps, it has quite a
different feeling when improvising. Reminds me of a starry sky.

This 55-tet twelve tone scale is actually
a close approximation to sixth comma meantone. It originates from
Tosi in 1723.

Each step of 55-tet is very close to a
syntonic comma in size (closest approximation of all is one step
of 56-tet).

The syntonic comma is the interval you
get if you go up four 3/2 major fifths, and compare this with the
result of going up a 5/4 major third plus two octaves. It is the
same as the interval between the E string of a violin and the
similarly pitched E high harmonic of the C string of a cello in a
string quartet if the players tune using pure instead of tempered
fifths, and is a little over 1/5th of a 12-tet semitone.

This makes a twelve tone scale
consisiting of two sizes of semitone, one of five, and one of
four approximate syntonic commas. Leopold Mozart (Wolfgang
Mozart's father) was in favour of this system and wrote a couple
of scales to be used as excerices for it. Teleman advocated it
for this same reason.

On page 4 he mentions that the tritone
sounds particularly good in sixth comma meantone, being at the
most optimal position in a certain sense, and this links in with
the improvisations - I remember noticing that the one from B to F
sounded particularly nice and enjoyed using it.

Interestingly, if one follows the logic
of the meantone scale itself, it has a syntonic comma of 0 (zero)
as Paul Erlich has just pointed out to me on MakeMicroMusic.

The fifth in 55-tet is 32 steps.

So going up four fifths as in c to e'',
means 128 steps. Going up two octaves is 110 steps. So the
difference between the two is 18 steps, which is the same as the
major third of 55-tet.

So what happens to the other major third
in the circle of fifths - the one that crosses the break in the
circle? (down 8 fifths)

Say, F# to Bb. One gets a sharper third
at 414.55 cents.

While in PYthagorean, the one going up
four fifths is the one that's sharp at 81/64 = 407.82, and the
one going down 8 fifths is flatter and near to the just
intonation 5/4 at 8192/6561 = 384.36 cents.

Improvisation in 7-tet
(seven tone equal temperament)

This uses the complete 7-tet "neutral
diminished seventh" consisiting of two of the 7-tet "neutral
diminished sevenths" stacked on top of each other to include
all of the notes of 7-tet in a seven note chord. You can use it
to modulate where you like just as with the 12-tet diminished
seventh, but more so.

More about the scale:

7-tet can be thought of as result of
stacking seven 11/9s on top of each other, and then tempering to
remove the Pythagorean comma. (11/9)^7 = two octaves + 31.86
cents - so you need to temper each 11/9 by -4.55 cents. It is
exactly analogous to stacking twelve 3/2s on top of each other to
get seven octaves plus 23.4 cents, then tempering each 3/2 by 2
cents to get the 12-tet equal temperament.

Tempering the 11/9s to 7-tet keeps them
reasonably in tune, but the (11/9)^2, which is fairly close to 3/2
at 694.8 cents instead of 702 cents, when tempered to 7-tet
becomes a 685.7 cents fifth - decidedly dissonant in most
timbres, with fast and prominent beating. It is especially so
when used as a triad with contrast of the relatively pure 11/9
neutral third (neutral = mid way between major and minor). I find
the 7-tet triad best thougth of as an incomplete "neutral
diminished seventh".

See also the 7-tet trio for violin, viola
and glockenspiel (with extra cello part in movement 2) on the
tunes page.

An octahedron also has four square cross-sections,
if you cut it through the middle. These are known as geodesic
squares (by analogy with the great circles on a globe). It is
well worth while to find these to help with hexany improvisations.

Each pair of identical intervals between
scale notes gives opposite sides of a square on the octahedron.
So one can see two of them straight away - the two 8/7 intervals
will make one of them, and in fact do so as 8/7 6/5 7/8 5/6, and
the two 7/6s make another one, as 7/6 5/4 6/7 4/5.

The last one is 7/5 3/2 5/7 2/3, which
takes more finding. In terms of scale degrees, the 7/5 is the
step from degrees 1 to 4. The 3/2 would then take one to degree 8
in the next octave, or if one looks for the same note in the
first octave, one goes down 4/3 from 4 to 2, instead of up 3/2
from 4 to 8.

So, this last square is 7/5 4/3 10/7 2/3,
and it joins the scale degrees 1 4 2 5 1

These squares are worth finding, as the
remaining two notes for each square then make all eight consonant
triads with the consecutive notes of the square.

The special thing about this scale is
that there is no special resting point or centre, as all the
triads and diads are points of rest. It is one of many scales of
this type invented by Erv Wilson, collectively known as
Combination Product Sets.

Though the hexany is a small scale with
only six notes, it is one that takes a lot of time to get used to
- if you have a try at composing in it, or improvising in it, one
needn't be disheartened if ones first attempts don't work at all.
It takes a fair while to get to know it. But it is well worth the
effort!

This is my first hexany improvisation
that has worked reasonably well, after many attempts. For my
hexany compositions, see Hexany recorder trio, Hexany phrase, but
they don't really exploit this wonderful weightless quality of
the scale.

Kraig Grady is the master of performance
and composition in CPS sets, with many years of experience of it.

E.g. the 1 3 7 11 hexany is the same
basic idea as the 1 3 5 7, but it has 11s instead of 5s. The
diads of the 1 3 5 7 hexany are 3/2 4/3 5/4 6/5 7/6 8/7, and the
diads of the 1 3 7 11 hexany are 3/2 4/3 7/6 8/7 11/8 16/11,
involving the numbers 3 7 and 11 instead of 3 5 and 7.

More elaborate CPS sets such as the
twenty note Eikosany have the same quality that every triad is a
point of rest, and have tetrads and pentands as well, and have
many component hexanies. So learning the various hexanies is a
first step towards exploring these more complex CPS sets.

Improvisation
in 17-tet diatonic

In Seventeen-tone Major scale = steps 3 3
1 3 3 3 1 in the system with seventeen equally spaced notes per
octave.

See 17-tet hurdy gurdy
player for more about it. This
scale is particularly festive and exciting, however can also have
a kind of subtle beauty and calm as well, and maybe some of that
may come across in this improvisation.